The Fourier Transform of Absolute Value of t

This page deals with the absolute value function, |t|. To start, we need to rewrite
the function k(t)=|t| and the sum of two other functions (
Now, since we know what the
Fourier Transform of the step function u(t) is, and we also know
what the Fourier Transform of a function times
Note that we use the following notation for the derivative of the dirac-delta impulse:
We can find the Fourier Transform of the second function (
Now, the question is, what is:
To determine this, recall that on the dirac-delta page,
we discussed how the dirac-delta impulse can be thought of as the limit of a sequence of functions becoming shorter and higher
with
Now, lets look at the derivative plus the "reflected" derivative (as in Eq [5]):
Since we know the limit as n goes to infinity of
Hence, we can get the solution, by combining Equations [2,4,8]:

*g(t)* and *h(t)*):
[Equation 1]
*t* is, we
can find the Fourier Transform of the first term in Equation [1]:
[Equation 2]
[Equation 3]
*h(t)*) in Equation [1] now:
[Equation 4]
[Equation 5]
*n*, for *n*=1,2,3,.... Now, if we look at the derivative of each one of these limiting functions (*fn*), we get:
[Equation 6]
[Equation 7]
*fn* is the dirac-delta impulse, we can get an answer to Equation [5]:
[Equation 8]
[Equation 9]

Fourier Transform Pairs Table of Contents